1. In triangle \( \triangle ABC \), if \( \angle ABC = 90^\circ \), \( AB = 6 \) cm, and \( BC = 8 \) cm, then what is the length of the circumradius of triangle \( \triangle ABC \)?

2. In triangle \(ABC\) and triangle \(DEF\), \(\angle ABC = \angle DEF\), \(\frac{AB}{DE} = \frac{BC}{EF}\), and the ratio of their areas is \(\frac{\triangle ABC}{\triangle DEF} = \frac{25}{16}\). If \(BC = 10\) cm, what is the length of \(EF\)?

3. In triangle \( \triangle ABC \), a line parallel to side BC intersects sides AB and AC at points P and Q respectively. If \( AB = 3 \times PB \) and \( BC = 18 \) cm, then what is the length of \( PQ \)?

(a) 10 cm (b) 9 cm (c) 12 cm (d) 8 cm

4. In triangle \(\triangle ABC\), AB = AC. Points E and F are the midpoints of sides AB and AC respectively. AD is perpendicular to BC, and AD = 4 cm. If EF = 3 cm, then what is the length of BD?

(a) 4 cm (b) 3 cm (c) 6 cm (d) 7 cm

5. In right-angled triangle ABC, \(\angle B = 90^\circ\), and M is the midpoint of the hypotenuse drawn from point B. If AB = 6 and AC = 8, then what is the length of BM?

(a) 10 (b) 4 (c) 16 (d) None of the above

6. In \(\triangle ABC\), \(\angle ABC = 90^\circ\) and \(BD \perp AC\); if \(AB = 6\) cm, \(BD = 3\) cm, and \(CD = 5.4\) cm, then calculate and write the length of side \(BC\).

7. In right-angled triangle \( \triangle ABC \), \( \angle ABC = 90^\circ \) and BD is perpendicular to AC. If BD = 8 cm and AD = 4 cm, what is the length of CD?

8. In triangle \( \triangle ABC \), if \( \angle B \) is a right angle and \( BC = \sqrt{3} \times AB \), then what is the value of \( \sin C \)?

(a) \(\frac{1}{2}\) (b) \(\frac{1}{\sqrt2}\) (c) \(\frac{\sqrt3}{2}\) (d) 1

9. In a circle centered at \(O\), two chords \(AB\) and \(CD\) have equal lengths. Given that \(\angle AOB = 60°\) and the radius of the circle is \(6\) cm, find the area of \(\triangle COD\).

(a) \(9\sqrt3\) square c.m (b) \(6\sqrt3\) square c.m (c) \(2\sqrt3\) square c.m (d) \(3\sqrt3\) square c.m

10. In \( \triangle ABC \), points \( D \) and \( E \) lie on sides \( AB \) and \( AC \) respectively, such that \( DE \parallel BC \) and \( AD:DB = 3:1 \). If \( EA = 33 \) cm, then the length of \( AC \) is _____.

11. In right-angled triangle \(\triangle ABC\), \(\angle ABC = 90^\circ\), AB = 3 cm, BC = 4 cm. If BD is perpendicular to AC, then find the length of BD.

12. In a right-angled triangle, the two acute angles are \(\theta\) and \(\phi\). If \( \tan\theta = \cfrac{5}{12} \), then what is the value of \( \sin\phi \)?

(a) \(\cfrac{12}{13}\) (b) \(\cfrac{5}{13}\) (c) \(\cfrac{1}{4}\) (d) \(\cfrac{10}{13}\)

13. In triangle ABC, the circumcenter is O; points A and B, C lie on opposite sides of the center. If \(\angle BOC = 120^\circ\), then what is the measure of \(\angle BAC\)?

(a) 50° (b) 60° (c) 70° (d) 80°

14. In triangle \(\triangle ABC\), O is the circumcenter. If \(\angle OAB = 35^\circ\), then what is the measure of \(\angle ACB\)?

(a) 45° (b) 50° (c) 55° (d) 60°

15. In a circle centered at \(O\), two equal chords \(AB\) and \(CD\) are given. If \(\angle AOB = 60^\circ\) and the radius of the circle is 6 cm, determine the area of \( \triangle COD \).

(a) \(6\sqrt3\) square cm (b) \(2\sqrt3\) square cm (c) \(2\sqrt3\) square cm (d) \(9\sqrt3\) square cm

16. If \(O\) is the circumcenter of \(\triangle ABC\) and \(\angle BAC = 50^\circ\), then find the measure of \(\angle OBC\).

(a) 100° (b) 30° (c) 40° (d) 50°

17. From vertex A of triangle \(\triangle ABC\), a perpendicular AD is drawn to the base BC, intersecting BC at point D. If \(AD^2 = BD \cdot CD\), prove that triangle ABC is a right-angled triangle and that \(\angle A = 90^\circ\).

18. Here’s the English translation of your math problem: If the ascending data set is \[ 27,\ 31,\ 46,\ 52,\ x,\ y + 2,\ 71,\ 79,\ 85,\ 90 \] and the median of the set is \(64\), then what is the value of \(x + y\)?

(a) 125 (b) 126 (c) 127 (d) 128

19. From an external point \(P\), two tangents \(PS\) and \(PT\) are drawn to a circle with center \(O\). \(QS\) is a chord of the circle that is parallel to \(PT\). If \(\angle SPT = 80^\circ\), then what is the measure of \(\angle QST\)?

20. In the circle with center \( O \), if \( AOB \) is a diameter and point \( C \) is on the circle such that \( AC = 3 \) cm and \( BC = 4 \) cm, what is the length of \( AB \)?

(a) 3 cm (b) 4 cm (c) 5 cm (d) 8 cm

21. In a circle with center \( O \), \( AB \) and \( CD \) are two equal-length chords. \( E \) is the midpoint of \( CD \), and \( \angle AOB = 70^\circ \). The value of angle \( \angle COE \) is:

(a) 70° (b) 110° (c) 35° (d) 55°

22. On triangle \( \triangle ABC \), points P and Q are such that \( \angle ABC = \angle APQ \). Given that \( AP = 3.6 \) cm, \( QC = 1.6 \) cm, and \( AQ = 4.8 \) cm, find the length of \( PB \).

(a) 1.2 cm (b) 2.4 cm (c) 6 cm (d) None of the above

23. In triangle \( \triangle ABC \), AD is a median. Point E divides AD in the ratio 1:2. The extended line BE intersects AC at point F. If \( AC = 10 \) cm, find the length of \( AF \).

(a) 5 cm (b) 4 cm (c) 2 cm (d) None of the above

24. Here’s the English translation of your question: **"If \( x = \frac{4ab}{a + b} \), then what is the value of \( \frac{x + 2a}{x - 2a} + \frac{x + 2b}{x - 2b} \)?"** Would you like me to solve it step by step?

(a) 1 (b) -2 (c) 2 (d) -1

25. Sure! Here's the English translation of your math question: **If** \(x = \frac{\sqrt{3}}{2}\), then what is the value of \[ \frac{\sqrt{1+x} + \sqrt{1-x}}{\sqrt{1+x} - \sqrt{1-x}}? \]

(a) \(2\sqrt3\) (b) \(\cfrac{1}{\sqrt3}\) (c) \(\sqrt3\) (d) \(\sqrt5\)

26. In right-angled triangle \( \triangle ABC \), where \( \angle A = 90^\circ \), a perpendicular \( AD \) is drawn from point \( A \) to the hypotenuse \( BC \). Prove that: \[ \frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle ACD} = \frac{BC^2}{AC^2} \]

27. In the adjacent figure, O is the center of the circle and BOA is the diameter. A tangent is drawn at point P on the circle, which intersects the extended line BA at point T. If \(\angle PBO = 30^\circ\), then what is the measure of \(\angle PTA\)?

28. In triangle \(\triangle ABC\), the incenter is \(O\), and \(\angle BOC = 120^\circ\). Find the measure of \(\angle BAC\).

29. In \(\triangle ABC\), the center of the incircle is \(O\), and the incircle touches the sides \(AB\), \(BC\), and \(CA\) at points \(P\), \(Q\), and \(R\) respectively. Given that \(AP = 4\) cm, \(BP = 6\) cm, \(AC = 12\) cm, and \(BC = x\) cm, determine the value of \(x\).

30. In \(\triangle ABC\), \(AB = 9\) cm, \(BC = 6\) cm, and \(CA = 7.5\) cm. In \(\triangle DEF\), the corresponding side to \(BC\) is \(EF\), and \(EF = 8\) cm. Given that \(\triangle ABC \sim \triangle DEF\), determine the perimeter of \(\triangle DEF\).